Signal-to-noise and spatial resolution in in-line imaging. 1. Basic theory, numerical simulations and planar experimental images.

Signal-to-noise ratio and spatial resolution are quantitatively analysed in the context of in-line (propagation based) X-ray phase-contrast imaging. It is known that free-space propagation of a coherent X-ray beam from the imaged object to the detector plane, followed by phase retrieval in accordance with Paganin's method, can increase the signal-to-noise in the resultant images without deteriorating the spatial resolution. This results in violation of the noise-resolution uncertainty principle and demonstrates `unreasonable' effectiveness of the method. On the other hand, when the process of free-space propagation is performed in software, using the detected intensity distribution in the object plane, it cannot reproduce the same effectiveness, due to the amplification of photon shot noise. Here, it is shown that the performance of Paganin's method is determined by just two dimensionless parameters: the Fresnel number and the ratio of the real decrement to the imaginary part of the refractive index of the imaged object. The relevant theoretical analysis is performed first, followed by computer simulations and then by a brief test using experimental images collected at a synchrotron beamline. More extensive experimental tests will be presented in the second part of this paper.

Young's double-slit interference with single hard X-ray photons.

Double-slit interference experiments using monochromatic hard X-rays with the energy of 25 keV are presented. The experiments were performed at a synchrotron source with a distance of 110 m between the interferometer and the detector to produce an interference pattern with a sufficiently broad period that could be adequately sampled by a photon-counting detector with 75 micrometre pixels. In the single-particle version of the experiment, over one million image frames with a single registered photon in each one were collected. The sum of these frames showed a clear presence of the interference pattern with the expected period. Subsequent analysis provided an objective estimation of the minimal number of detected photons required to determine, in accordance with the Rose criterion, the presence of the photon interference. Apart from a general theoretical interest, these investigations were aimed at exploring the possibility of medical X-ray phase-contrast imaging in photon-counting regime at minimal radiation doses.

Three-dimensional contrast-transfer-function approach in phase-contrast tomography.

A new method is developed for 3D reconstruction of multimaterial objects using propagation-based x-ray phase-contrast tomography (PB-CT) with phase retrieval via contrast-transfer-function (CTF) formalism. The approach differs from conventional PB-CT algorithms, which apply phase retrieval to individual 2D projections. Instead, this method involves performing phase retrieval to the CT-reconstructed volume in 3D. The CTF formalism is further extended to the cases of partially coherent illumination and strongly absorbing samples. Simulated results demonstrate that the proposed post-reconstruction CTF method provides fast and stable phase retrieval, producing results equivalent to conventional pre-reconstruction 2D CTF phase retrieval. Moreover, it is shown that application can be highly localized to isolated objects of interest, without a significant loss of quality, thus leading to increased computational efficiency. Combined with the extended validity of the CTF to greater propagation distances, this method provides additional advantages over approaches based on the transport-of-intensity equation.

Unified fast reconstruction algorithm for conventional, phase-contrast, and diffraction tomography.

A unified method for three-dimensional reconstruction of objects from transmission images collected at multiple illumination directions is described. The method may be applicable to experimental conditions relevant to absorption-based, phase-contrast, or diffraction imaging using x rays, electrons, and other forms of penetrating radiation or matter waves. Both the phase retrieval (also known as contrast transfer function correction) and the effect of Ewald sphere curvature (in the cases with a shallow depth of field and significant in-object diffraction) are incorporated in the proposed algorithm and can be taken into account. Multiple scattering is not treated explicitly but can be mitigated as a result of angular averaging that constitutes an essential feature of the method. The corresponding numerical algorithm is based on three-dimensional gridding which allows for fast computational implementation, including a straightforward parallelization. The algorithm can be used with any scanning geometry involving plane-wave illumination. A software code implementing the proposed algorithm has been developed, tested on simulated and experimental image data, and made publicly available.

Method for virtual optical sectioning and tomography utilizing shallow depth of field.

A method is proposed for high-resolution, three-dimensional reconstruction of internal structures of objects from planar transmission images. The described approach can be used with any form of radiation or matter waves, in principle, provided that the depth of field is smaller than the thickness of the sample. The physical optics basis for the method is elucidated, and the reconstruction algorithm is presented in detail. A simulated example demonstrates an application of the method to three-dimensional electron transmission imaging of a nanoparticle under realistic radiation dose and spatial resolution constraints. It is envisaged that the method can be applicable in high-resolution transmission electron microscopy, soft x-ray microscopy, ultrasound imaging, and other areas.

A Method for High-Resolution Three-Dimensional Reconstruction with Ewald Sphere Curvature Correction from Transmission Electron Images.

A method for three-dimensional reconstruction of objects from defocused images collected at multiple illumination directions in high-resolution transmission electron microscopy is presented. The method effectively corrects for the Ewald sphere curvature by taking into account the in-particle propagation of the electron beam. Numerical simulations demonstrate that the proposed method is capable of accurately reconstructing biological molecules or nanoparticles from high-resolution defocused images under conditions achievable in single-particle electron cryo-microscopy or electron tomography with realistic radiation doses, non-trivial aberrations, multiple scattering, and other experimentally relevant factors. The physics of the method is based on the well-known Diffraction Tomography formalism, but with the phase-retrieval step modified to include a conjugation of the phase (i.e., multiplication of the phase by a negative constant). At each illumination direction, numerically backpropagating the beam with the conjugated phase produces maximum contrast at the location of individual atoms in the molecule or nanoparticle. The resultant algorithm, Conjugated Holographic Reconstruction, can potentially be incorporated into established software tools for single-particle analysis, such as, for example, RELION or FREALIGN, in place of the conventional contrast transfer function correction procedure, in order to account for the Ewald sphere curvature and improve the spatial resolution of the three-dimensional reconstruction.

Effect of x-ray energy on the radiological image quality in propagation-based phase-contrast computed tomography of the breast.

Purpose: Breast cancer is the most common cancer in women in developing and developed countries and is responsible for 15% of women's cancer deaths worldwide. Conventional absorption-based breast imaging techniques lack sufficient contrast for comprehensive diagnosis. Propagation-based phase-contrast computed tomography (PB-CT) is a developing technique that exploits a more contrast-sensitive property of x-rays: x-ray refraction. X-ray absorption, refraction, and contrast-to-noise in the corresponding images depend on the x-ray energy used, for the same/fixed radiation dose. The aim of this paper is to explore the relationship between x-ray energy and radiological image quality in PB-CT imaging. Approach: Thirty-nine mastectomy samples were scanned at the imaging and medical beamline at the Australian Synchrotron. Samples were scanned at various x-ray energies of 26, 28, 30, 32, 34, and 60 keV using a Hamamatsu Flat Panel detector at the same object-to-detector distance of 6 m and mean glandular dose of 4 mGy. A total of 132 image sets were produced for analysis. Seven observers rated PB-CT images against absorption-based CT (AB-CT) images of the same samples on a five-point scale. A visual grading characteristics (VGC) study was used to determine the difference in image quality. Results: PB-CT images produced at 28, 30, 32, and 34 keV x-ray energies demonstrated statistically significant higher image quality than reference AB-CT images. The optimum x-ray energy, 30 keV, displayed the largest area under the curve ( AUC VGC ) of 0.754 ( p = 0.009 ). This was followed by 32 keV ( AUC VGC = 0.731 , p ≤ 0.001 ), 34 keV ( AUC VGC = 0.723 , p ≤ 0.001 ), and 28 keV ( AUC VGC = 0.654 , p = 0.015 ). Conclusions: An optimum energy range (around 30 keV) in the PB-CT technique allows for higher image quality at a dose comparable to conventional mammographic techniques. This results in improved radiological image quality compared with conventional techniques, which may ultimately lead to higher diagnostic efficacy and a reduction in breast cancer mortalities.

Propagation-Based Phase-Contrast CT of the Breast Demonstrates Higher Quality Than Conventional Absorption-Based CT Even at Lower Radiation Dose.

RATIONALE AND OBJECTIVES: Propagation-based phase-contrast CT (PB-CT) is an advanced X-ray imaging technology that exploits both refraction and absorption of the transmitted X-ray beam. This study was aimed at optimizing the experimental conditions of PB-CT for breast cancer imaging and examined its performance relative to conventional absorption-based CT (AB-CT) in terms of image quality and radiation dose. MATERIALS AND METHODS: Surgically excised breast mastectomy specimens (n = 12) were scanned using both PB-CT and AB-CT techniques under varying imaging conditions. To evaluate the radiological image quality, visual grading characteristics (VGC) analysis was used in which 11 breast specialist radiologists compared the overall image quality of PB-CT images with respect to the corresponding AB-CT images. The area under the VGC curve was calculated to measure the differences between PB-CT and AB-CT images. RESULTS: The highest radiological quality was obtained for PB-CT images using a 32 keV energy X-ray beam and by applying the Homogeneous Transport of Intensity Equation phase retrieval with the value of its parameter γ set to one-half of the theoretically optimal value for the given materials. Using these optimized conditions, the image quality of PB-CT images obtained at 4 mGy and 2 mGy mean glandular dose was significantly higher than AB-CT images at 4 mGy (AUCVGC = 0.901, p = 0.001 and AUCVGC = 0.819, p = 0.011, respectively). CONCLUSION: PB-CT achieves a higher radiological image quality compared to AB-CT even at a considerably lower mean glandular dose. Successful translation of the PB-CT technique for breast cancer imaging can potentially result in improved breast cancer diagnosis.

Recovery of undamaged electron-density maps in the presence of damage-induced partial coherence in single-particle imaging.

Resolving the electronic structure of single biological molecules in their native state was among the primary motivations behind X-ray free-electron lasers. The ultra-short pulses they produce can outrun the atomic motion induced by radiation damage, but the electronic structure of the sample is still significantly modified from its original state. This paper explores the decoherence of the scattered signal induced by temporal evolution of the electronic structure in the sample molecule. It is shown that the undamaged electron density of a single-molecule sample can often be retrieved using only the two most occupied modes from the coherent mode decomposition of the partially coherent diffraction fluence.

Effect of radiation damage and illumination variability on signal-to-noise ratio in X-ray free-electron laser single-particle imaging.

The deterioration of both the signal-to-noise ratio and the spatial resolution in the electron-density distribution reconstructed from diffraction intensities collected at different orientations of a sample is analysed theoretically with respect to the radiation damage to the sample and the variations in the X-ray intensities illuminating different copies of the sample. The simple analytical expressions and numerical estimates obtained for models of radiation damage and incident X-ray pulses may be helpful in planning X-ray free-electron laser (XFEL) imaging experiments and in analysis of experimental data. This approach to the analysis of partially coherent X-ray imaging configurations can potentially be used for analysis of other forms of imaging where the temporal behaviour of the sample and the incident intensity during exposure may affect the inverse problem of sample reconstruction.

Noise-resolution uncertainty principle in classical and quantum systems.

We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the "classical" noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.

Comparison of propagation-based CT using synchrotron radiation and conventional cone-beam CT for breast imaging.

OBJECTIVES: To evaluate and compare the image quality of propagation-based phase-contrast computed tomography (PB-CT) using synchrotron radiation and conventional cone-beam breast computed tomography (CBBCT) based on various radiological image quality criteria. METHODS: Eight excised breast tissue samples of various sizes and containing different lesion types were scanned using PB-CT at a synchrotron facility and using CBBCT at a university-affiliated breast imaging centre. PB-CT scans were performed at two different mean glandular dose (MGD) levels: standard (5.8 mGy) and low (1.5 mGy), for comparison with CBBCT scans at the standard MGD (5.8 mGy). Image quality assessment was carried out using six quality criteria and six independent medical imaging experts in a reading room with mammography workstations. The interobserver agreement between readers was evaluated using intraclass correlation coefficient (ICC), and image quality was compared between the two breast imaging modalities using the area under the visual grading characteristic curve (AUCVGC). RESULTS: Interobserver agreement between the readers showed moderate reliability for five image criteria (ICC: ranging from 0.488 to 0.633) and low reliability for one criterion (image noise) (ICC 0.307). For five image quality criteria (overall quality, perceptible contrast, lesion sharpness, normal tissue interfaces, and calcification visibility), both standard-dose PB-CT images (AUCVGC 0.958 to 1, p ≤ .05) and low dose PB-CT images (AUCVGC 0.785 to 0.834, p ≤ .05) were of significantly higher image quality than standard-dose CBBCT images. CONCLUSIONS: Synchrotron-based PB-CT can achieve a significantly higher radiological image quality at a substantially lower radiation dose compared with conventional CBBCT. KEY POINTS: • PB-CT using synchrotron radiation results in higher image quality than conventional CBBCT for breast imaging. • PB-CT using synchrotron radiation requires a lower radiation dose than conventional CBBCT for breast imaging. • PB-CT can help clinicians diagnose patients with breast cancer.

Fast three-dimensional phase retrieval in propagation-based X-ray tomography.

The following article describes a method for 3D reconstruction of multi-material objects based on propagation-based X-ray phase-contrast tomography (PB-CT) with phase retrieval using the homogeneous form of the transport of intensity equation (TIE-Hom). Unlike conventional PB-CT algorithms that perform phase retrieval of individual projections, the described post-reconstruction phase-retrieval method is applied in 3D to a localized region of the CT-reconstructed volume. This work demonstrates, via numerical simulations, the accuracy and noise characteristics of the method under a variety of experimental conditions, comparing it with both conventional absorption tomography and 2D TIE-Hom phase retrieval applied to projection images. The results indicate that the 3D post-reconstruction method generally achieves a modest improvement in noise suppression over existing PB-CT methods. It is also shown that potentially large computational gains over projection-based phase retrieval for multi-material samples are possible. In particular, constraining phase retrieval to a localized 3D region of interest reduces the overall computational cost and eliminates the need for multiple CT reconstructions and global 2D phase retrieval operations for each material within the sample.

Toward Improving Breast Cancer Imaging: Radiological Assessment of Propagation-Based Phase-Contrast CT Technology.

RATIONALE AND OBJECTIVES: This study employs clinical/radiological evaluation in establishing the optimum imaging conditions for breast cancer imaging using the X-ray propagation-based phase-contrast tomography. MATERIALS AND METHODS: Two series of experiments were conducted and in total 161 synchrotron-based computed tomography (CT) reconstructions of one breast mastectomy specimen were produced at different imaging conditions. Imaging factors include sample-to-detector distance, X-ray energy, CT reconstruction method, phase retrieval algorithm applied to the CT projection images and maximum intensity projection. Observers including breast radiologists and medical imaging experts compared the quality of the reconstructed images with reference images approximating the conventional (absorption) CT. Various radiological image quality attributes in a visual grading analysis design were used for the radiological assessments. RESULTS: The results show that the application of the longest achievable sample-to-detector distance (9.31 m), the lowest employed X-ray energy (32 keV), the full phase retrieval, and the maximum intensity projection can significantly improve the radiological quality of the image. Several combinations of imaging variables resulted in images with very high-quality scores. CONCLUSION: The results of the present study will support future experimental and clinical attempts to further optimize this innovative approach to breast cancer imaging.

Signal-to-noise, spatial resolution and information capacity of coherent diffraction imaging.

It is shown that the average signal-to-noise ratio (SNR) in the three-dimensional electron-density distribution of a sample reconstructed by coherent diffractive imaging cannot exceed twice the square root of the ratio of the mean total number of scattered photons detected during the scan and the number of spatially resolved voxels in the reconstructed volume. This result leads to an upper bound on Shannon's information capacity of this imaging method by specifying the maximum number of distinguishable density distributions within the reconstructed volume when the radiation dose delivered to the sample and the spatial resolution are both fixed. If the spatially averaged SNR in the reconstructed electron density is fixed instead, the radiation dose is shown to be proportional to the third or fourth power of the spatial resolution, depending on the sampling of the three-dimensional diffraction space and the scattering power of the sample.

X-Ray Phase-Contrast Technology in Breast Imaging: Principles, Options, and Clinical Application.

OBJECTIVE: The purpose of this article is to review different x-ray phase-contrast breast imaging techniques and their potential application in clinical settings. CONCLUSION: Phase-contrast imaging depicts not only the absorption contrast but also the refraction contrast of the transmitted x-ray beam. Early data suggest that this new modality may overcome some of the diagnostic limitations associated with current clinically available mammography systems and that it has potential for improving breast cancer detection.

On the "unreasonable" effectiveness of transport of intensity imaging and optical deconvolution.

The effectiveness of reconstructive imaging using the homogeneous transport of intensity equation may be regarded as "unreasonable," because it has been shown to significantly increase signal-to-noise ratio while preserving spatial resolution, compared to equivalent conventional absorption-based imaging techniques at the same photon fluence. We reconcile this surprising behavior by analyzing the propagation of noise in typical in-line holography experiments. This analysis indicates that novel imaging techniques may be designed that produce high signal-to-noise images at low radiation doses without sacrificing spatial resolution.

CT dose reduction factors in the thousands using X-ray phase contrast.

Phase-contrast X-ray imaging can improve the visibility of weakly absorbing objects (e.g. soft tissues) by an order of magnitude or more compared to conventional radiographs. Combining phase retrieval with computed tomography (CT) can increase the signal-to-noise ratio (SNR) by up to two orders of magnitude over conventional CT at the same radiation dose, without loss of image quality. Our experiments reveal that as the radiation dose decreases, the relative improvement in SNR increases. We show that this enhancement can be traded for a reduction in dose greater than the square of the gain in SNR. Upon reducing the dose 300 fold, the phase-retrieved SNR was still up to 9.6 ± 0.2 times larger than the absorption contrast data with spatial resolution in the tens of microns. We show that this theoretically reveals the potential for dose reduction factors in the tens of thousands without loss in image quality, which would have a profound impact on medical and industrial imaging applications.

On the van Cittert-Zernike theorem for intensity correlations and its applications.

A reciprocal relationship between the autocovariance of the light intensity in the source plane and in the far-field detector plane is presented in a form analogous to the classical van Cittert-Zernike theorem, but involving intensity correlation functions. A "classical" version of the reciprocity relationship is considered first, based on the assumption of circular Gaussian statistics of the complex amplitudes in the source plane. The result is consistent with the theory of Hanbury Brown-Twiss interferometry, but it is shown to be also applicable to estimation of the source size or the spatial resolution of the detector from the noise power spectrum of flat-field images. An alternative version of the van Cittert-Zernike theorem for intensity correlations is then derived for a quantized electromagnetic beam in a coherent state, which leads to Poisson statistics for the intrinsic intensity of the beam.

On noise-resolution uncertainty in quantum field theory.

An uncertainty inequality is presented that establishes a lower limit for the product of the variance of the time-averaged intensity of a mode of a quantized electromagnetic field and the degree of its spatial localization. The lower limit is determined by the vacuum fluctuations within the volume corresponding to the width of the mode. This result also leads to a generalized form of the Heisenberg uncertainty principle for boson fields in which the lower limit for the product of uncertainties in the spatial and momentum localization of a mode is equal to the product of Planck's constant and a dimensionless functional which reflects the joint signal-to-noise ratio of the position and momentum of vacuum fluctuations in the region of the phase space occupied by the mode. Experimental X-ray synchrotron measurements provide an initial verification of the proposed theory in the case of Poisson statistics.